LMFDB - Embedded newform 9522.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9522,2,Mod(1,9522)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9522, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9522.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.284630$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.06731 q^{5} -4.42518 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.06731 q^{5} -4.42518 q^{7} -1.00000 q^{8} +1.06731 q^{10} -1.64964 q^{11} +5.47112 q^{13} +4.42518 q^{14} +1.00000 q^{16} +3.92613 q^{17} -6.60149 q^{19} -1.06731 q^{20} +1.64964 q^{22} -3.86085 q^{25} -5.47112 q^{26} -4.42518 q^{28} +6.39946 q^{29} -3.01491 q^{31} -1.00000 q^{32} -3.92613 q^{34} +4.72304 q^{35} +2.21076 q^{37} +6.60149 q^{38} +1.06731 q^{40} -4.09177 q^{41} +2.91325 q^{43} -1.64964 q^{44} +9.62306 q^{47} +12.5822 q^{49} +3.86085 q^{50} +5.47112 q^{52} -2.23040 q^{53} +1.76068 q^{55} +4.42518 q^{56} -6.39946 q^{58} -9.75208 q^{59} +1.55946 q^{61} +3.01491 q^{62} +1.00000 q^{64} -5.83938 q^{65} -12.0453 q^{67} +3.92613 q^{68} -4.72304 q^{70} +5.26416 q^{71} +7.18556 q^{73} -2.21076 q^{74} -6.60149 q^{76} +7.29997 q^{77} -2.69381 q^{79} -1.06731 q^{80} +4.09177 q^{82} +5.53843 q^{83} -4.19039 q^{85} -2.91325 q^{86} +1.64964 q^{88} +11.3261 q^{89} -24.2107 q^{91} -9.62306 q^{94} +7.04583 q^{95} -11.9794 q^{97} -12.5822 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - q^{5} - 11 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 - q^5 - 11 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} - q^{5} - 11 q^{7} - 5 q^{8} + q^{10} + 11 q^{11} + 12 q^{13} + 11 q^{14} + 5 q^{16} - q^{17} - 15 q^{19} - q^{20} - 11 q^{22} + 6 q^{25} - 12 q^{26} - 11 q^{28} - q^{29} - 18 q^{31} - 5 q^{32} + q^{34} + 11 q^{35} - 10 q^{37} + 15 q^{38} + q^{40} + 16 q^{41} - 18 q^{43} + 11 q^{44} + 4 q^{47} + 20 q^{49} - 6 q^{50} + 12 q^{52} - q^{53} - 22 q^{55} + 11 q^{56} + q^{58} - 2 q^{59} + q^{61} + 18 q^{62} + 5 q^{64} + 24 q^{65} - 29 q^{67} - q^{68} - 11 q^{70} + 11 q^{71} - 8 q^{73} + 10 q^{74} - 15 q^{76} - 11 q^{77} - 40 q^{79} - q^{80} - 16 q^{82} + 8 q^{83} - 13 q^{85} + 18 q^{86} - 11 q^{88} + 2 q^{89} - 33 q^{91} - 4 q^{94} + 3 q^{95} - 17 q^{97} - 20 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - q^5 - 11 * q^7 - 5 * q^8 + q^10 + 11 * q^11 + 12 * q^13 + 11 * q^14 + 5 * q^16 - q^17 - 15 * q^19 - q^20 - 11 * q^22 + 6 * q^25 - 12 * q^26 - 11 * q^28 - q^29 - 18 * q^31 - 5 * q^32 + q^34 + 11 * q^35 - 10 * q^37 + 15 * q^38 + q^40 + 16 * q^41 - 18 * q^43 + 11 * q^44 + 4 * q^47 + 20 * q^49 - 6 * q^50 + 12 * q^52 - q^53 - 22 * q^55 + 11 * q^56 + q^58 - 2 * q^59 + q^61 + 18 * q^62 + 5 * q^64 + 24 * q^65 - 29 * q^67 - q^68 - 11 * q^70 + 11 * q^71 - 8 * q^73 + 10 * q^74 - 15 * q^76 - 11 * q^77 - 40 * q^79 - q^80 - 16 * q^82 + 8 * q^83 - 13 * q^85 + 18 * q^86 - 11 * q^88 + 2 * q^89 - 33 * q^91 - 4 * q^94 + 3 * q^95 - 17 * q^97 - 20 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.06731	−0.477315	−0.238658	−	0.971104i	$-0.576707\pi$
$5$	−1.06731	−0.477315	−0.238658	+	0.971104i	$0.576707\pi$
$6$	0	0
$7$	−4.42518	−1.67256	−0.836281	−	0.548302i	$-0.815275\pi$
$7$	−4.42518	−1.67256	−0.836281	+	0.548302i	$0.815275\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.06731	0.337513
$11$	−1.64964	−0.497386	−0.248693	−	0.968582i	$-0.580001\pi$
$11$	−1.64964	−0.497386	−0.248693	+	0.968582i	$0.580001\pi$
$12$	0	0
$13$	5.47112	1.51742	0.758708	−	0.651431i	$-0.225831\pi$
$13$	5.47112	1.51742	0.758708	+	0.651431i	$0.225831\pi$
$14$	4.42518	1.18268
$15$	0	0
$16$	1.00000	0.250000
$17$	3.92613	0.952226	0.476113	−	0.879384i	$-0.342045\pi$
$17$	3.92613	0.952226	0.476113	+	0.879384i	$0.342045\pi$
$18$	0	0
$19$	−6.60149	−1.51449	−0.757243	−	0.653133i	$-0.773454\pi$
$19$	−6.60149	−1.51449	−0.757243	+	0.653133i	$0.773454\pi$
$20$	−1.06731	−0.238658
$21$	0	0
$22$	1.64964	0.351705
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.86085	−0.772170
$26$	−5.47112	−1.07297
$27$	0	0
$28$	−4.42518	−0.836281
$29$	6.39946	1.18835	0.594175	−	0.804336i	$-0.297479\pi$
$29$	6.39946	1.18835	0.594175	+	0.804336i	$0.297479\pi$
$30$	0	0
$31$	−3.01491	−0.541494	−0.270747	−	0.962650i	$-0.587271\pi$
$31$	−3.01491	−0.541494	−0.270747	+	0.962650i	$0.587271\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.92613	−0.673325
$35$	4.72304	0.798339
$36$	0	0
$37$	2.21076	0.363446	0.181723	−	0.983350i	$-0.441833\pi$
$37$	2.21076	0.363446	0.181723	+	0.983350i	$0.441833\pi$
$38$	6.60149	1.07090
$39$	0	0
$40$	1.06731	0.168756
$41$	−4.09177	−0.639027	−0.319514	−	0.947582i	$-0.603520\pi$
$41$	−4.09177	−0.639027	−0.319514	+	0.947582i	$0.603520\pi$
$42$	0	0
$43$	2.91325	0.444266	0.222133	−	0.975016i	$-0.428698\pi$
$43$	2.91325	0.444266	0.222133	+	0.975016i	$0.428698\pi$
$44$	−1.64964	−0.248693
$45$	0	0
$46$	0	0
$47$	9.62306	1.40367	0.701834	−	0.712341i	$-0.252365\pi$
$47$	9.62306	1.40367	0.701834	+	0.712341i	$0.252365\pi$
$48$	0	0
$49$	12.5822	1.79746
$50$	3.86085	0.546007
$51$	0	0
$52$	5.47112	0.758708
$53$	−2.23040	−0.306368	−0.153184	−	0.988198i	$-0.548953\pi$
$53$	−2.23040	−0.306368	−0.153184	+	0.988198i	$0.548953\pi$
$54$	0	0
$55$	1.76068	0.237410
$56$	4.42518	0.591340
$57$	0	0
$58$	−6.39946	−0.840290
$59$	−9.75208	−1.26961	−0.634807	−	0.772671i	$-0.718920\pi$
$59$	−9.75208	−1.26961	−0.634807	+	0.772671i	$0.718920\pi$
$60$	0	0
$61$	1.55946	0.199668	0.0998339	−	0.995004i	$-0.468169\pi$
$61$	1.55946	0.199668	0.0998339	+	0.995004i	$0.468169\pi$
$62$	3.01491	0.382894
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.83938	−0.724285
$66$	0	0
$67$	−12.0453	−1.47157	−0.735784	−	0.677216i	$-0.763186\pi$
$67$	−12.0453	−1.47157	−0.735784	+	0.677216i	$0.763186\pi$
$68$	3.92613	0.476113
$69$	0	0
$70$	−4.72304	−0.564511
$71$	5.26416	0.624740	0.312370	−	0.949960i	$-0.398877\pi$
$71$	5.26416	0.624740	0.312370	+	0.949960i	$0.398877\pi$
$72$	0	0
$73$	7.18556	0.841006	0.420503	−	0.907291i	$-0.361854\pi$
$73$	7.18556	0.841006	0.420503	+	0.907291i	$0.361854\pi$
$74$	−2.21076	−0.256995
$75$	0	0
$76$	−6.60149	−0.757243
$77$	7.29997	0.831909
$78$	0	0
$79$	−2.69381	−0.303077	−0.151538	−	0.988451i	$-0.548423\pi$
$79$	−2.69381	−0.303077	−0.151538	+	0.988451i	$0.548423\pi$
$80$	−1.06731	−0.119329
$81$	0	0
$82$	4.09177	0.451861
$83$	5.53843	0.607922	0.303961	−	0.952685i	$-0.401691\pi$
$83$	5.53843	0.607922	0.303961	+	0.952685i	$0.401691\pi$
$84$	0	0
$85$	−4.19039	−0.454512
$86$	−2.91325	−0.314144
$87$	0	0
$88$	1.64964	0.175853
$89$	11.3261	1.20057	0.600283	−	0.799788i	$-0.295055\pi$
$89$	11.3261	1.20057	0.600283	+	0.799788i	$0.295055\pi$
$90$	0	0
$91$	−24.2107	−2.53797
$92$	0	0
$93$	0	0
$94$	−9.62306	−0.992543
$95$	7.04583	0.722887
$96$	0	0
$97$	−11.9794	−1.21633	−0.608163	−	0.793812i	$-0.708093\pi$
$97$	−11.9794	−1.21633	−0.608163	+	0.793812i	$0.708093\pi$
$98$	−12.5822	−1.27100
$99$	0	0
$100$	−3.86085	−0.386085
$101$	18.4610	1.83693	0.918467	−	0.395498i	$-0.129428\pi$
$101$	18.4610	1.83693	0.918467	+	0.395498i	$0.129428\pi$
$102$	0	0
$103$	11.4277	1.12601	0.563003	−	0.826455i	$-0.309646\pi$
$103$	11.4277	1.12601	0.563003	+	0.826455i	$0.309646\pi$
$104$	−5.47112	−0.536487
$105$	0	0
$106$	2.23040	0.216635
$107$	14.8641	1.43697	0.718485	−	0.695543i	$-0.244836\pi$
$107$	14.8641	1.43697	0.718485	+	0.695543i	$0.244836\pi$
$108$	0	0
$109$	−12.2830	−1.17650	−0.588249	−	0.808680i	$-0.700182\pi$
$109$	−12.2830	−1.17650	−0.588249	+	0.808680i	$0.700182\pi$
$110$	−1.76068	−0.167874
$111$	0	0
$112$	−4.42518	−0.418140
$113$	−3.01462	−0.283591	−0.141796	−	0.989896i	$-0.545288\pi$
$113$	−3.01462	−0.283591	−0.141796	+	0.989896i	$0.545288\pi$
$114$	0	0
$115$	0	0
$116$	6.39946	0.594175
$117$	0	0
$118$	9.75208	0.897752
$119$	−17.3738	−1.59266
$120$	0	0
$121$	−8.27868	−0.752607
$122$	−1.55946	−0.141186
$123$	0	0
$124$	−3.01491	−0.270747
$125$	9.45727	0.845884
$126$	0	0
$127$	−5.45241	−0.483823	−0.241912	−	0.970298i	$-0.577774\pi$
$127$	−5.45241	−0.483823	−0.241912	+	0.970298i	$0.577774\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	5.83938	0.512147
$131$	8.67749	0.758156	0.379078	−	0.925365i	$-0.376241\pi$
$131$	8.67749	0.758156	0.379078	+	0.925365i	$0.376241\pi$
$132$	0	0
$133$	29.2128	2.53307
$134$	12.0453	1.04056
$135$	0	0
$136$	−3.92613	−0.336663
$137$	8.98175	0.767363	0.383682	−	0.923465i	$-0.374656\pi$
$137$	8.98175	0.767363	0.383682	+	0.923465i	$0.374656\pi$
$138$	0	0
$139$	3.06287	0.259789	0.129895	−	0.991528i	$-0.458536\pi$
$139$	3.06287	0.259789	0.129895	+	0.991528i	$0.458536\pi$
$140$	4.72304	0.399169
$141$	0	0
$142$	−5.26416	−0.441758
$143$	−9.02540	−0.754742
$144$	0	0
$145$	−6.83020	−0.567217
$146$	−7.18556	−0.594681
$147$	0	0
$148$	2.21076	0.181723
$149$	−20.4130	−1.67230	−0.836149	−	0.548502i	$-0.815198\pi$
$149$	−20.4130	−1.67230	−0.836149	+	0.548502i	$0.815198\pi$
$150$	0	0
$151$	−12.7143	−1.03468	−0.517339	−	0.855780i	$-0.673078\pi$
$151$	−12.7143	−1.03468	−0.517339	+	0.855780i	$0.673078\pi$
$152$	6.60149	0.535452
$153$	0	0
$154$	−7.29997	−0.588249
$155$	3.21784	0.258463
$156$	0	0
$157$	1.14480	0.0913647	0.0456823	−	0.998956i	$-0.485454\pi$
$157$	1.14480	0.0913647	0.0456823	+	0.998956i	$0.485454\pi$
$158$	2.69381	0.214308
$159$	0	0
$160$	1.06731	0.0843782
$161$	0	0
$162$	0	0
$163$	4.80338	0.376230	0.188115	−	0.982147i	$-0.439762\pi$
$163$	4.80338	0.376230	0.188115	+	0.982147i	$0.439762\pi$
$164$	−4.09177	−0.319514
$165$	0	0
$166$	−5.53843	−0.429865
$167$	−11.7479	−0.909078	−0.454539	−	0.890727i	$-0.650196\pi$
$167$	−11.7479	−0.909078	−0.454539	+	0.890727i	$0.650196\pi$
$168$	0	0
$169$	16.9332	1.30255
$170$	4.19039	0.321388
$171$	0	0
$172$	2.91325	0.222133
$173$	5.11919	0.389205	0.194602	−	0.980882i	$-0.437658\pi$
$173$	5.11919	0.389205	0.194602	+	0.980882i	$0.437658\pi$
$174$	0	0
$175$	17.0850	1.29150
$176$	−1.64964	−0.124347
$177$	0	0
$178$	−11.3261	−0.848928
$179$	18.7474	1.40124	0.700622	−	0.713533i	$-0.252906\pi$
$179$	18.7474	1.40124	0.700622	+	0.713533i	$0.252906\pi$
$180$	0	0
$181$	7.26568	0.540053	0.270027	−	0.962853i	$-0.412967\pi$
$181$	7.26568	0.540053	0.270027	+	0.962853i	$0.412967\pi$
$182$	24.2107	1.79462
$183$	0	0
$184$	0	0
$185$	−2.35956	−0.173478
$186$	0	0
$187$	−6.47671	−0.473624
$188$	9.62306	0.701834
$189$	0	0
$190$	−7.04583	−0.511158
$191$	25.1816	1.82208	0.911040	−	0.412319i	$-0.135281\pi$
$191$	25.1816	1.82208	0.911040	+	0.412319i	$0.135281\pi$
$192$	0	0
$193$	16.9067	1.21697	0.608486	−	0.793564i	$-0.291777\pi$
$193$	16.9067	1.21697	0.608486	+	0.793564i	$0.291777\pi$
$194$	11.9794	0.860072
$195$	0	0
$196$	12.5822	0.898731
$197$	12.8420	0.914952	0.457476	−	0.889222i	$-0.348754\pi$
$197$	12.8420	0.914952	0.457476	+	0.889222i	$0.348754\pi$
$198$	0	0
$199$	−13.6046	−0.964402	−0.482201	−	0.876061i	$-0.660162\pi$
$199$	−13.6046	−0.964402	−0.482201	+	0.876061i	$0.660162\pi$
$200$	3.86085	0.273003
$201$	0	0
$202$	−18.4610	−1.29891
$203$	−28.3188	−1.98759
$204$	0	0
$205$	4.36718	0.305017
$206$	−11.4277	−0.796207
$207$	0	0
$208$	5.47112	0.379354
$209$	10.8901	0.753285
$210$	0	0
$211$	−21.1423	−1.45549	−0.727746	−	0.685846i	$-0.759432\pi$
$211$	−21.1423	−1.45549	−0.727746	+	0.685846i	$0.759432\pi$
$212$	−2.23040	−0.153184
$213$	0	0
$214$	−14.8641	−1.01609
$215$	−3.10933	−0.212055
$216$	0	0
$217$	13.3415	0.905683
$218$	12.2830	0.831909
$219$	0	0
$220$	1.76068	0.118705
$221$	21.4803	1.44492
$222$	0	0
$223$	−2.55189	−0.170887	−0.0854435	−	0.996343i	$-0.527231\pi$
$223$	−2.55189	−0.170887	−0.0854435	+	0.996343i	$0.527231\pi$
$224$	4.42518	0.295670
$225$	0	0
$226$	3.01462	0.200529
$227$	−19.7228	−1.30905	−0.654524	−	0.756041i	$-0.727131\pi$
$227$	−19.7228	−1.30905	−0.654524	+	0.756041i	$0.727131\pi$
$228$	0	0
$229$	11.2054	0.740473	0.370237	−	0.928937i	$-0.379277\pi$
$229$	11.2054	0.740473	0.370237	+	0.928937i	$0.379277\pi$
$230$	0	0
$231$	0	0
$232$	−6.39946	−0.420145
$233$	−18.0191	−1.18047	−0.590236	−	0.807230i	$-0.700966\pi$
$233$	−18.0191	−1.18047	−0.590236	+	0.807230i	$0.700966\pi$
$234$	0	0
$235$	−10.2708	−0.669992
$236$	−9.75208	−0.634807
$237$	0	0
$238$	17.3738	1.12618
$239$	−27.2479	−1.76252	−0.881261	−	0.472630i	$-0.843305\pi$
$239$	−27.2479	−1.76252	−0.881261	+	0.472630i	$0.843305\pi$
$240$	0	0
$241$	22.2765	1.43496	0.717478	−	0.696581i	$-0.245296\pi$
$241$	22.2765	1.43496	0.717478	+	0.696581i	$0.245296\pi$
$242$	8.27868	0.532173
$243$	0	0
$244$	1.55946	0.0998339
$245$	−13.4291	−0.857955
$246$	0	0
$247$	−36.1176	−2.29811
$248$	3.01491	0.191447
$249$	0	0
$250$	−9.45727	−0.598130
$251$	−10.1994	−0.643779	−0.321889	−	0.946777i	$-0.604318\pi$
$251$	−10.1994	−0.643779	−0.321889	+	0.946777i	$0.604318\pi$
$252$	0	0
$253$	0	0
$254$	5.45241	0.342115
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.8386	−1.73652	−0.868261	−	0.496108i	$-0.834762\pi$
$257$	−27.8386	−1.73652	−0.868261	+	0.496108i	$0.834762\pi$
$258$	0	0
$259$	−9.78300	−0.607886
$260$	−5.83938	−0.362143
$261$	0	0
$262$	−8.67749	−0.536097
$263$	7.54866	0.465470	0.232735	−	0.972540i	$-0.425232\pi$
$263$	7.54866	0.465470	0.232735	+	0.972540i	$0.425232\pi$
$264$	0	0
$265$	2.38052	0.146234
$266$	−29.2128	−1.79115
$267$	0	0
$268$	−12.0453	−0.735784
$269$	−5.87690	−0.358321	−0.179160	−	0.983820i	$-0.557338\pi$
$269$	−5.87690	−0.358321	−0.179160	+	0.983820i	$0.557338\pi$
$270$	0	0
$271$	21.2053	1.28813	0.644065	−	0.764971i	$-0.277247\pi$
$271$	21.2053	1.28813	0.644065	+	0.764971i	$0.277247\pi$
$272$	3.92613	0.238056
$273$	0	0
$274$	−8.98175	−0.542608
$275$	6.36903	0.384067
$276$	0	0
$277$	3.17305	0.190650	0.0953251	−	0.995446i	$-0.469611\pi$
$277$	3.17305	0.190650	0.0953251	+	0.995446i	$0.469611\pi$
$278$	−3.06287	−0.183699
$279$	0	0
$280$	−4.72304	−0.282255
$281$	−18.2707	−1.08994	−0.544970	−	0.838456i	$-0.683459\pi$
$281$	−18.2707	−1.08994	−0.544970	+	0.838456i	$0.683459\pi$
$282$	0	0
$283$	−22.5002	−1.33750	−0.668750	−	0.743488i	$-0.733170\pi$
$283$	−22.5002	−1.33750	−0.668750	+	0.743488i	$0.733170\pi$
$284$	5.26416	0.312370
$285$	0	0
$286$	9.02540	0.533683
$287$	18.1068	1.06881
$288$	0	0
$289$	−1.58552	−0.0932659
$290$	6.83020	0.401083
$291$	0	0
$292$	7.18556	0.420503
$293$	−10.4017	−0.607677	−0.303838	−	0.952724i	$-0.598268\pi$
$293$	−10.4017	−0.607677	−0.303838	+	0.952724i	$0.598268\pi$
$294$	0	0
$295$	10.4085	0.606005
$296$	−2.21076	−0.128498
$297$	0	0
$298$	20.4130	1.18249
$299$	0	0
$300$	0	0
$301$	−12.8916	−0.743062
$302$	12.7143	0.731628
$303$	0	0
$304$	−6.60149	−0.378622
$305$	−1.66442	−0.0953044
$306$	0	0
$307$	5.13212	0.292905	0.146453	−	0.989218i	$-0.453214\pi$
$307$	5.13212	0.292905	0.146453	+	0.989218i	$0.453214\pi$
$308$	7.29997	0.415955
$309$	0	0
$310$	−3.21784	−0.182761
$311$	−21.3709	−1.21183	−0.605917	−	0.795528i	$-0.707194\pi$
$311$	−21.3709	−1.21183	−0.605917	+	0.795528i	$0.707194\pi$
$312$	0	0
$313$	−15.9653	−0.902411	−0.451205	−	0.892420i	$-0.649006\pi$
$313$	−15.9653	−0.902411	−0.451205	+	0.892420i	$0.649006\pi$
$314$	−1.14480	−0.0646046
$315$	0	0
$316$	−2.69381	−0.151538
$317$	−10.8838	−0.611297	−0.305649	−	0.952144i	$-0.598873\pi$
$317$	−10.8838	−0.611297	−0.305649	+	0.952144i	$0.598873\pi$
$318$	0	0
$319$	−10.5568	−0.591069
$320$	−1.06731	−0.0596644
$321$	0	0
$322$	0	0
$323$	−25.9183	−1.44213
$324$	0	0
$325$	−21.1232	−1.17170
$326$	−4.80338	−0.266035
$327$	0	0
$328$	4.09177	0.225930
$329$	−42.5838	−2.34772
$330$	0	0
$331$	−13.5079	−0.742459	−0.371230	−	0.928541i	$-0.621064\pi$
$331$	−13.5079	−0.742459	−0.371230	+	0.928541i	$0.621064\pi$
$332$	5.53843	0.303961
$333$	0	0
$334$	11.7479	0.642815
$335$	12.8561	0.702402
$336$	0	0
$337$	1.90607	0.103830	0.0519152	−	0.998651i	$-0.483467\pi$
$337$	1.90607	0.103830	0.0519152	+	0.998651i	$0.483467\pi$
$338$	−16.9332	−0.921042
$339$	0	0
$340$	−4.19039	−0.227256
$341$	4.97353	0.269332
$342$	0	0
$343$	−24.7024	−1.33380
$344$	−2.91325	−0.157072
$345$	0	0
$346$	−5.11919	−0.275209
$347$	−9.62944	−0.516936	−0.258468	−	0.966020i	$-0.583218\pi$
$347$	−9.62944	−0.516936	−0.258468	+	0.966020i	$0.583218\pi$
$348$	0	0
$349$	−31.7107	−1.69744	−0.848719	−	0.528845i	$-0.822625\pi$
$349$	−31.7107	−1.69744	−0.848719	+	0.528845i	$0.822625\pi$
$350$	−17.0850	−0.913230
$351$	0	0
$352$	1.64964	0.0879263
$353$	−7.73095	−0.411477	−0.205738	−	0.978607i	$-0.565960\pi$
$353$	−7.73095	−0.411477	−0.205738	+	0.978607i	$0.565960\pi$
$354$	0	0
$355$	−5.61848	−0.298198
$356$	11.3261	0.600283
$357$	0	0
$358$	−18.7474	−0.990829
$359$	−37.0366	−1.95472	−0.977359	−	0.211586i	$-0.932137\pi$
$359$	−37.0366	−1.95472	−0.977359	+	0.211586i	$0.932137\pi$
$360$	0	0
$361$	24.5797	1.29367
$362$	−7.26568	−0.381875
$363$	0	0
$364$	−24.2107	−1.26899
$365$	−7.66921	−0.401425
$366$	0	0
$367$	12.1890	0.636260	0.318130	−	0.948047i	$-0.396945\pi$
$367$	12.1890	0.636260	0.318130	+	0.948047i	$0.396945\pi$
$368$	0	0
$369$	0	0
$370$	2.35956	0.122668
$371$	9.86991	0.512420
$372$	0	0
$373$	−5.54618	−0.287170	−0.143585	−	0.989638i	$-0.545863\pi$
$373$	−5.54618	−0.287170	−0.143585	+	0.989638i	$0.545863\pi$
$374$	6.47671	0.334903
$375$	0	0
$376$	−9.62306	−0.496271
$377$	35.0122	1.80322
$378$	0	0
$379$	10.2328	0.525626	0.262813	−	0.964847i	$-0.415350\pi$
$379$	10.2328	0.525626	0.262813	+	0.964847i	$0.415350\pi$
$380$	7.04583	0.361444
$381$	0	0
$382$	−25.1816	−1.28840
$383$	−4.49082	−0.229470	−0.114735	−	0.993396i	$-0.536602\pi$
$383$	−4.49082	−0.229470	−0.114735	+	0.993396i	$0.536602\pi$
$384$	0	0
$385$	−7.79133	−0.397083
$386$	−16.9067	−0.860530
$387$	0	0
$388$	−11.9794	−0.608163
$389$	3.41829	0.173314	0.0866571	−	0.996238i	$-0.472382\pi$
$389$	3.41829	0.173314	0.0866571	+	0.996238i	$0.472382\pi$
$390$	0	0
$391$	0	0
$392$	−12.5822	−0.635499
$393$	0	0
$394$	−12.8420	−0.646969
$395$	2.87512	0.144663
$396$	0	0
$397$	−25.0502	−1.25723	−0.628617	−	0.777715i	$-0.716379\pi$
$397$	−25.0502	−1.25723	−0.628617	+	0.777715i	$0.716379\pi$
$398$	13.6046	0.681935
$399$	0	0
$400$	−3.86085	−0.193043
$401$	−8.95685	−0.447284	−0.223642	−	0.974671i	$-0.571795\pi$
$401$	−8.95685	−0.447284	−0.223642	+	0.974671i	$0.571795\pi$
$402$	0	0
$403$	−16.4950	−0.821672
$404$	18.4610	0.918467
$405$	0	0
$406$	28.3188	1.40544
$407$	−3.64696	−0.180773
$408$	0	0
$409$	−23.2815	−1.15120	−0.575599	−	0.817732i	$-0.695231\pi$
$409$	−23.2815	−1.15120	−0.575599	+	0.817732i	$0.695231\pi$
$410$	−4.36718	−0.215680
$411$	0	0
$412$	11.4277	0.563003
$413$	43.1547	2.12351
$414$	0	0
$415$	−5.91121	−0.290170
$416$	−5.47112	−0.268244
$417$	0	0
$418$	−10.8901	−0.532653
$419$	−2.24964	−0.109902	−0.0549510	−	0.998489i	$-0.517500\pi$
$419$	−2.24964	−0.109902	−0.0549510	+	0.998489i	$0.517500\pi$
$420$	0	0
$421$	18.6618	0.909521	0.454761	−	0.890614i	$-0.349725\pi$
$421$	18.6618	0.909521	0.454761	+	0.890614i	$0.349725\pi$
$422$	21.1423	1.02919
$423$	0	0
$424$	2.23040	0.108318
$425$	−15.1582	−0.735281
$426$	0	0
$427$	−6.90087	−0.333957
$428$	14.8641	0.718485
$429$	0	0
$430$	3.10933	0.149945
$431$	4.81723	0.232038	0.116019	−	0.993247i	$-0.462987\pi$
$431$	4.81723	0.232038	0.116019	+	0.993247i	$0.462987\pi$
$432$	0	0
$433$	6.25528	0.300609	0.150305	−	0.988640i	$-0.451975\pi$
$433$	6.25528	0.300609	0.150305	+	0.988640i	$0.451975\pi$
$434$	−13.3415	−0.640414
$435$	0	0
$436$	−12.2830	−0.588249
$437$	0	0
$438$	0	0
$439$	−13.5454	−0.646486	−0.323243	−	0.946316i	$-0.604773\pi$
$439$	−13.5454	−0.646486	−0.323243	+	0.946316i	$0.604773\pi$
$440$	−1.76068	−0.0839371
$441$	0	0
$442$	−21.4803	−1.02171
$443$	−5.53987	−0.263207	−0.131603	−	0.991302i	$-0.542013\pi$
$443$	−5.53987	−0.263207	−0.131603	+	0.991302i	$0.542013\pi$
$444$	0	0
$445$	−12.0885	−0.573048
$446$	2.55189	0.120835
$447$	0	0
$448$	−4.42518	−0.209070
$449$	13.5471	0.639327	0.319664	−	0.947531i	$-0.396430\pi$
$449$	13.5471	0.639327	0.319664	+	0.947531i	$0.396430\pi$
$450$	0	0
$451$	6.74996	0.317843
$452$	−3.01462	−0.141796
$453$	0	0
$454$	19.7228	0.925637
$455$	25.8403	1.21141
$456$	0	0
$457$	11.1137	0.519877	0.259939	−	0.965625i	$-0.416298\pi$
$457$	11.1137	0.519877	0.259939	+	0.965625i	$0.416298\pi$
$458$	−11.2054	−0.523594
$459$	0	0
$460$	0	0
$461$	33.3324	1.55245	0.776223	−	0.630459i	$-0.217133\pi$
$461$	33.3324	1.55245	0.776223	+	0.630459i	$0.217133\pi$
$462$	0	0
$463$	−19.4351	−0.903228	−0.451614	−	0.892213i	$-0.649152\pi$
$463$	−19.4351	−0.903228	−0.451614	+	0.892213i	$0.649152\pi$
$464$	6.39946	0.297087
$465$	0	0
$466$	18.0191	0.834720
$467$	−33.5250	−1.55135	−0.775676	−	0.631131i	$-0.782591\pi$
$467$	−33.5250	−1.55135	−0.775676	+	0.631131i	$0.782591\pi$
$468$	0	0
$469$	53.3027	2.46129
$470$	10.2708	0.473756
$471$	0	0
$472$	9.75208	0.448876
$473$	−4.80582	−0.220972
$474$	0	0
$475$	25.4874	1.16944
$476$	−17.3738	−0.796328
$477$	0	0
$478$	27.2479	1.24629
$479$	−1.65573	−0.0756521	−0.0378260	−	0.999284i	$-0.512043\pi$
$479$	−1.65573	−0.0756521	−0.0378260	+	0.999284i	$0.512043\pi$
$480$	0	0
$481$	12.0953	0.551499
$482$	−22.2765	−1.01467
$483$	0	0
$484$	−8.27868	−0.376303
$485$	12.7857	0.580571
$486$	0	0
$487$	−20.9954	−0.951394	−0.475697	−	0.879609i	$-0.657804\pi$
$487$	−20.9954	−0.951394	−0.475697	+	0.879609i	$0.657804\pi$
$488$	−1.55946	−0.0705932
$489$	0	0
$490$	13.4291	0.606666
$491$	−25.8128	−1.16492	−0.582458	−	0.812861i	$-0.697909\pi$
$491$	−25.8128	−1.16492	−0.582458	+	0.812861i	$0.697909\pi$
$492$	0	0
$493$	25.1251	1.13158
$494$	36.1176	1.62501
$495$	0	0
$496$	−3.01491	−0.135374
$497$	−23.2948	−1.04492
$498$	0	0
$499$	−6.43766	−0.288189	−0.144095	−	0.989564i	$-0.546027\pi$
$499$	−6.43766	−0.288189	−0.144095	+	0.989564i	$0.546027\pi$
$500$	9.45727	0.422942
$501$	0	0
$502$	10.1994	0.455220
$503$	10.1631	0.453151	0.226576	−	0.973994i	$-0.427247\pi$
$503$	10.1631	0.453151	0.226576	+	0.973994i	$0.427247\pi$
$504$	0	0
$505$	−19.7035	−0.876796
$506$	0	0
$507$	0	0
$508$	−5.45241	−0.241912
$509$	26.2279	1.16253	0.581266	−	0.813714i	$-0.302558\pi$
$509$	26.2279	1.16253	0.581266	+	0.813714i	$0.302558\pi$
$510$	0	0
$511$	−31.7974	−1.40663
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	27.8386	1.22791
$515$	−12.1969	−0.537460
$516$	0	0
$517$	−15.8746	−0.698165
$518$	9.78300	0.429840
$519$	0	0
$520$	5.83938	0.256074
$521$	15.4025	0.674798	0.337399	−	0.941362i	$-0.390453\pi$
$521$	15.4025	0.674798	0.337399	+	0.941362i	$0.390453\pi$
$522$	0	0
$523$	26.2241	1.14670	0.573349	−	0.819311i	$-0.305644\pi$
$523$	26.2241	1.14670	0.573349	+	0.819311i	$0.305644\pi$
$524$	8.67749	0.379078
$525$	0	0
$526$	−7.54866	−0.329137
$527$	−11.8369	−0.515625
$528$	0	0
$529$	0	0
$530$	−2.38052	−0.103403
$531$	0	0
$532$	29.2128	1.26654
$533$	−22.3866	−0.969670
$534$	0	0
$535$	−15.8646	−0.685887
$536$	12.0453	0.520278
$537$	0	0
$538$	5.87690	0.253371
$539$	−20.7562	−0.894033
$540$	0	0
$541$	0.996978	0.0428634	0.0214317	−	0.999770i	$-0.493178\pi$
$541$	0.996978	0.0428634	0.0214317	+	0.999770i	$0.493178\pi$
$542$	−21.2053	−0.910845
$543$	0	0
$544$	−3.92613	−0.168331
$545$	13.1098	0.561560
$546$	0	0
$547$	5.83245	0.249377	0.124689	−	0.992196i	$-0.460207\pi$
$547$	5.83245	0.249377	0.124689	+	0.992196i	$0.460207\pi$
$548$	8.98175	0.383682
$549$	0	0
$550$	−6.36903	−0.271576
$551$	−42.2460	−1.79974
$552$	0	0
$553$	11.9206	0.506914
$554$	−3.17305	−0.134810
$555$	0	0
$556$	3.06287	0.129895
$557$	−32.8498	−1.39189	−0.695946	−	0.718094i	$-0.745015\pi$
$557$	−32.8498	−1.39189	−0.695946	+	0.718094i	$0.745015\pi$
$558$	0	0
$559$	15.9387	0.674136
$560$	4.72304	0.199585
$561$	0	0
$562$	18.2707	0.770704
$563$	−12.0434	−0.507568	−0.253784	−	0.967261i	$-0.581675\pi$
$563$	−12.0434	−0.507568	−0.253784	+	0.967261i	$0.581675\pi$
$564$	0	0
$565$	3.21753	0.135362
$566$	22.5002	0.945755
$567$	0	0
$568$	−5.26416	−0.220879
$569$	−12.9218	−0.541708	−0.270854	−	0.962620i	$-0.587306\pi$
$569$	−12.9218	−0.541708	−0.270854	+	0.962620i	$0.587306\pi$
$570$	0	0
$571$	−33.8981	−1.41859	−0.709296	−	0.704911i	$-0.750987\pi$
$571$	−33.8981	−1.41859	−0.709296	+	0.704911i	$0.750987\pi$
$572$	−9.02540	−0.377371
$573$	0	0
$574$	−18.1068	−0.755764
$575$	0	0
$576$	0	0
$577$	−22.4539	−0.934768	−0.467384	−	0.884054i	$-0.654803\pi$
$577$	−22.4539	−0.934768	−0.467384	+	0.884054i	$0.654803\pi$
$578$	1.58552	0.0659490
$579$	0	0
$580$	−6.83020	−0.283609
$581$	−24.5086	−1.01679
$582$	0	0
$583$	3.67936	0.152383
$584$	−7.18556	−0.297341
$585$	0	0
$586$	10.4017	0.429692
$587$	1.15068	0.0474935	0.0237467	−	0.999718i	$-0.492440\pi$
$587$	1.15068	0.0474935	0.0237467	+	0.999718i	$0.492440\pi$
$588$	0	0
$589$	19.9029	0.820086
$590$	−10.4085	−0.428511
$591$	0	0
$592$	2.21076	0.0908616
$593$	3.28119	0.134742	0.0673712	−	0.997728i	$-0.478539\pi$
$593$	3.28119	0.134742	0.0673712	+	0.997728i	$0.478539\pi$
$594$	0	0
$595$	18.5432	0.760199
$596$	−20.4130	−0.836149
$597$	0	0
$598$	0	0
$599$	−3.76746	−0.153934	−0.0769671	−	0.997034i	$-0.524524\pi$
$599$	−3.76746	−0.153934	−0.0769671	+	0.997034i	$0.524524\pi$
$600$	0	0
$601$	19.6834	0.802904	0.401452	−	0.915880i	$-0.368506\pi$
$601$	19.6834	0.802904	0.401452	+	0.915880i	$0.368506\pi$
$602$	12.8916	0.525424
$603$	0	0
$604$	−12.7143	−0.517339
$605$	8.83590	0.359231
$606$	0	0
$607$	42.9460	1.74313	0.871563	−	0.490284i	$-0.163107\pi$
$607$	42.9460	1.74313	0.871563	+	0.490284i	$0.163107\pi$
$608$	6.60149	0.267726
$609$	0	0
$610$	1.66442	0.0673904
$611$	52.6489	2.12995
$612$	0	0
$613$	−23.3854	−0.944526	−0.472263	−	0.881458i	$-0.656563\pi$
$613$	−23.3854	−0.944526	−0.472263	+	0.881458i	$0.656563\pi$
$614$	−5.13212	−0.207115
$615$	0	0
$616$	−7.29997	−0.294124
$617$	7.46322	0.300458	0.150229	−	0.988651i	$-0.451999\pi$
$617$	7.46322	0.300458	0.150229	+	0.988651i	$0.451999\pi$
$618$	0	0
$619$	−32.5795	−1.30948	−0.654741	−	0.755853i	$-0.727222\pi$
$619$	−32.5795	−1.30948	−0.654741	+	0.755853i	$0.727222\pi$
$620$	3.21784	0.129232
$621$	0	0
$622$	21.3709	0.856896
$623$	−50.1201	−2.00802
$624$	0	0
$625$	9.21043	0.368417
$626$	15.9653	0.638101
$627$	0	0
$628$	1.14480	0.0456823
$629$	8.67972	0.346083
$630$	0	0
$631$	17.2415	0.686374	0.343187	−	0.939267i	$-0.388494\pi$
$631$	17.2415	0.686374	0.343187	+	0.939267i	$0.388494\pi$
$632$	2.69381	0.107154
$633$	0	0
$634$	10.8838	0.432252
$635$	5.81941	0.230936
$636$	0	0
$637$	68.8389	2.72750
$638$	10.5568	0.417949
$639$	0	0
$640$	1.06731	0.0421891
$641$	40.6615	1.60603	0.803017	−	0.595957i	$-0.203227\pi$
$641$	40.6615	1.60603	0.803017	+	0.595957i	$0.203227\pi$
$642$	0	0
$643$	11.4111	0.450008	0.225004	−	0.974358i	$-0.427760\pi$
$643$	11.4111	0.450008	0.225004	+	0.974358i	$0.427760\pi$
$644$	0	0
$645$	0	0
$646$	25.9183	1.01974
$647$	−34.5625	−1.35879	−0.679396	−	0.733772i	$-0.737758\pi$
$647$	−34.5625	−1.35879	−0.679396	+	0.733772i	$0.737758\pi$
$648$	0	0
$649$	16.0875	0.631488
$650$	21.1232	0.828519
$651$	0	0
$652$	4.80338	0.188115
$653$	−8.57392	−0.335524	−0.167762	−	0.985828i	$-0.553654\pi$
$653$	−8.57392	−0.335524	−0.167762	+	0.985828i	$0.553654\pi$
$654$	0	0
$655$	−9.26156	−0.361879
$656$	−4.09177	−0.159757
$657$	0	0
$658$	42.5838	1.66009
$659$	−4.22348	−0.164523	−0.0822616	−	0.996611i	$-0.526214\pi$
$659$	−4.22348	−0.164523	−0.0822616	+	0.996611i	$0.526214\pi$
$660$	0	0
$661$	23.1140	0.899029	0.449514	−	0.893273i	$-0.351597\pi$
$661$	23.1140	0.899029	0.449514	+	0.893273i	$0.351597\pi$
$662$	13.5079	0.524998
$663$	0	0
$664$	−5.53843	−0.214933
$665$	−31.1791	−1.20907
$666$	0	0
$667$	0	0
$668$	−11.7479	−0.454539
$669$	0	0
$670$	−12.8561	−0.496673
$671$	−2.57255	−0.0993120
$672$	0	0
$673$	−45.0158	−1.73523	−0.867616	−	0.497235i	$-0.834349\pi$
$673$	−45.0158	−1.73523	−0.867616	+	0.497235i	$0.834349\pi$
$674$	−1.90607	−0.0734192
$675$	0	0
$676$	16.9332	0.651275
$677$	−31.0382	−1.19290	−0.596448	−	0.802652i	$-0.703422\pi$
$677$	−31.0382	−1.19290	−0.596448	+	0.802652i	$0.703422\pi$
$678$	0	0
$679$	53.0111	2.03438
$680$	4.19039	0.160694
$681$	0	0
$682$	−4.97353	−0.190446
$683$	25.9817	0.994164	0.497082	−	0.867704i	$-0.334405\pi$
$683$	25.9817	0.994164	0.497082	+	0.867704i	$0.334405\pi$
$684$	0	0
$685$	−9.58631	−0.366274
$686$	24.7024	0.943141
$687$	0	0
$688$	2.91325	0.111067
$689$	−12.2028	−0.464888
$690$	0	0
$691$	33.5110	1.27482	0.637410	−	0.770525i	$-0.280006\pi$
$691$	33.5110	1.27482	0.637410	+	0.770525i	$0.280006\pi$
$692$	5.11919	0.194602
$693$	0	0
$694$	9.62944	0.365529
$695$	−3.26903	−0.124001
$696$	0	0
$697$	−16.0648	−0.608498
$698$	31.7107	1.20027
$699$	0	0
$700$	17.0850	0.645751
$701$	−5.85943	−0.221308	−0.110654	−	0.993859i	$-0.535294\pi$
$701$	−5.85943	−0.221308	−0.110654	+	0.993859i	$0.535294\pi$
$702$	0	0
$703$	−14.5943	−0.550434
$704$	−1.64964	−0.0621733
$705$	0	0
$706$	7.73095	0.290958
$707$	−81.6931	−3.07238
$708$	0	0
$709$	−45.0960	−1.69362	−0.846808	−	0.531899i	$-0.821479\pi$
$709$	−45.0960	−1.69362	−0.846808	+	0.531899i	$0.821479\pi$
$710$	5.61848	0.210858
$711$	0	0
$712$	−11.3261	−0.424464
$713$	0	0
$714$	0	0
$715$	9.63289	0.360250
$716$	18.7474	0.700622
$717$	0	0
$718$	37.0366	1.38219
$719$	32.9210	1.22775	0.613874	−	0.789404i	$-0.289610\pi$
$719$	32.9210	1.22775	0.613874	+	0.789404i	$0.289610\pi$
$720$	0	0
$721$	−50.5697	−1.88331
$722$	−24.5797	−0.914762
$723$	0	0
$724$	7.26568	0.270027
$725$	−24.7074	−0.917609
$726$	0	0
$727$	2.50599	0.0929421	0.0464710	−	0.998920i	$-0.485202\pi$
$727$	2.50599	0.0929421	0.0464710	+	0.998920i	$0.485202\pi$
$728$	24.2107	0.897308
$729$	0	0
$730$	7.66921	0.283850
$731$	11.4378	0.423042
$732$	0	0
$733$	11.0964	0.409856	0.204928	−	0.978777i	$-0.434304\pi$
$733$	11.0964	0.409856	0.204928	+	0.978777i	$0.434304\pi$
$734$	−12.1890	−0.449903
$735$	0	0
$736$	0	0
$737$	19.8705	0.731938
$738$	0	0
$739$	−36.1487	−1.32975	−0.664876	−	0.746954i	$-0.731515\pi$
$739$	−36.1487	−1.32975	−0.664876	+	0.746954i	$0.731515\pi$
$740$	−2.35956	−0.0867392
$741$	0	0
$742$	−9.86991	−0.362336
$743$	26.5639	0.974534	0.487267	−	0.873253i	$-0.337994\pi$
$743$	26.5639	0.974534	0.487267	+	0.873253i	$0.337994\pi$
$744$	0	0
$745$	21.7870	0.798213
$746$	5.54618	0.203060
$747$	0	0
$748$	−6.47671	−0.236812
$749$	−65.7764	−2.40342
$750$	0	0
$751$	−13.7898	−0.503198	−0.251599	−	0.967832i	$-0.580956\pi$
$751$	−13.7898	−0.503198	−0.251599	+	0.967832i	$0.580956\pi$
$752$	9.62306	0.350917
$753$	0	0
$754$	−35.0122	−1.27507
$755$	13.5701	0.493868
$756$	0	0
$757$	38.1039	1.38491	0.692455	−	0.721461i	$-0.256529\pi$
$757$	38.1039	1.38491	0.692455	+	0.721461i	$0.256529\pi$
$758$	−10.2328	−0.371674
$759$	0	0
$760$	−7.04583	−0.255579
$761$	−43.2965	−1.56950	−0.784748	−	0.619815i	$-0.787208\pi$
$761$	−43.2965	−1.56950	−0.784748	+	0.619815i	$0.787208\pi$
$762$	0	0
$763$	54.3545	1.96776
$764$	25.1816	0.911040
$765$	0	0
$766$	4.49082	0.162260
$767$	−53.3548	−1.92653
$768$	0	0
$769$	35.0733	1.26477	0.632387	−	0.774652i	$-0.282075\pi$
$769$	35.0733	1.26477	0.632387	+	0.774652i	$0.282075\pi$
$770$	7.79133	0.280780
$771$	0	0
$772$	16.9067	0.608486
$773$	−14.8799	−0.535192	−0.267596	−	0.963531i	$-0.586229\pi$
$773$	−14.8799	−0.535192	−0.267596	+	0.963531i	$0.586229\pi$
$774$	0	0
$775$	11.6401	0.418126
$776$	11.9794	0.430036
$777$	0	0
$778$	−3.41829	−0.122552
$779$	27.0118	0.967798
$780$	0	0
$781$	−8.68398	−0.310737
$782$	0	0
$783$	0	0
$784$	12.5822	0.449365
$785$	−1.22185	−0.0436097
$786$	0	0
$787$	−36.5315	−1.30221	−0.651103	−	0.758989i	$-0.725694\pi$
$787$	−36.5315	−1.30221	−0.651103	+	0.758989i	$0.725694\pi$
$788$	12.8420	0.457476
$789$	0	0
$790$	−2.87512	−0.102292
$791$	13.3402	0.474324
$792$	0	0
$793$	8.53197	0.302979
$794$	25.0502	0.888998
$795$	0	0
$796$	−13.6046	−0.482201
$797$	−45.5877	−1.61480	−0.807399	−	0.590006i	$-0.799125\pi$
$797$	−45.5877	−1.61480	−0.807399	+	0.590006i	$0.799125\pi$
$798$	0	0
$799$	37.7814	1.33661
$800$	3.86085	0.136502
$801$	0	0
$802$	8.95685	0.316277
$803$	−11.8536	−0.418305
$804$	0	0
$805$	0	0
$806$	16.4950	0.581010
$807$	0	0
$808$	−18.4610	−0.649454
$809$	21.8252	0.767332	0.383666	−	0.923472i	$-0.374661\pi$
$809$	21.8252	0.767332	0.383666	+	0.923472i	$0.374661\pi$
$810$	0	0
$811$	30.3605	1.06610	0.533050	−	0.846084i	$-0.321046\pi$
$811$	30.3605	1.06610	0.533050	+	0.846084i	$0.321046\pi$
$812$	−28.3188	−0.993794
$813$	0	0
$814$	3.64696	0.127826
$815$	−5.12669	−0.179580
$816$	0	0
$817$	−19.2318	−0.672835
$818$	23.2815	0.814020
$819$	0	0
$820$	4.36718	0.152509
$821$	34.0023	1.18669	0.593344	−	0.804949i	$-0.297808\pi$
$821$	34.0023	1.18669	0.593344	+	0.804949i	$0.297808\pi$
$822$	0	0
$823$	52.3590	1.82512	0.912561	−	0.408941i	$-0.134102\pi$
$823$	52.3590	1.82512	0.912561	+	0.408941i	$0.134102\pi$
$824$	−11.4277	−0.398103
$825$	0	0
$826$	−43.1547	−1.50155
$827$	−20.0689	−0.697864	−0.348932	−	0.937148i	$-0.613456\pi$
$827$	−20.0689	−0.697864	−0.348932	+	0.937148i	$0.613456\pi$
$828$	0	0
$829$	−30.6227	−1.06357	−0.531785	−	0.846879i	$-0.678479\pi$
$829$	−30.6227	−1.06357	−0.531785	+	0.846879i	$0.678479\pi$
$830$	5.91121	0.205181
$831$	0	0
$832$	5.47112	0.189677
$833$	49.3994	1.71159
$834$	0	0
$835$	12.5386	0.433916
$836$	10.8901	0.376642
$837$	0	0
$838$	2.24964	0.0777125
$839$	39.9432	1.37899	0.689497	−	0.724289i	$-0.257832\pi$
$839$	39.9432	1.37899	0.689497	+	0.724289i	$0.257832\pi$
$840$	0	0
$841$	11.9531	0.412175
$842$	−18.6618	−0.643129
$843$	0	0
$844$	−21.1423	−0.727746
$845$	−18.0729	−0.621727
$846$	0	0
$847$	36.6346	1.25878
$848$	−2.23040	−0.0765921
$849$	0	0
$850$	15.1582	0.519922
$851$	0	0
$852$	0	0
$853$	−3.87282	−0.132603	−0.0663014	−	0.997800i	$-0.521120\pi$
$853$	−3.87282	−0.132603	−0.0663014	+	0.997800i	$0.521120\pi$
$854$	6.90087	0.236143
$855$	0	0
$856$	−14.8641	−0.508045
$857$	−7.20505	−0.246120	−0.123060	−	0.992399i	$-0.539271\pi$
$857$	−7.20505	−0.246120	−0.123060	+	0.992399i	$0.539271\pi$
$858$	0	0
$859$	10.2462	0.349595	0.174797	−	0.984604i	$-0.444073\pi$
$859$	10.2462	0.349595	0.174797	+	0.984604i	$0.444073\pi$
$860$	−3.10933	−0.106027
$861$	0	0
$862$	−4.81723	−0.164076
$863$	−37.1420	−1.26433	−0.632164	−	0.774835i	$-0.717833\pi$
$863$	−37.1420	−1.26433	−0.632164	+	0.774835i	$0.717833\pi$
$864$	0	0
$865$	−5.46376	−0.185773
$866$	−6.25528	−0.212563
$867$	0	0
$868$	13.3415	0.452841
$869$	4.44382	0.150746
$870$	0	0
$871$	−65.9013	−2.23298
$872$	12.2830	0.415955
$873$	0	0
$874$	0	0
$875$	−41.8501	−1.41479
$876$	0	0
$877$	−1.43016	−0.0482930	−0.0241465	−	0.999708i	$-0.507687\pi$
$877$	−1.43016	−0.0482930	−0.0241465	+	0.999708i	$0.507687\pi$
$878$	13.5454	0.457134
$879$	0	0
$880$	1.76068	0.0593525
$881$	−19.8790	−0.669740	−0.334870	−	0.942264i	$-0.608692\pi$
$881$	−19.8790	−0.669740	−0.334870	+	0.942264i	$0.608692\pi$
$882$	0	0
$883$	16.2884	0.548147	0.274073	−	0.961709i	$-0.411629\pi$
$883$	16.2884	0.548147	0.274073	+	0.961709i	$0.411629\pi$
$884$	21.4803	0.722461
$885$	0	0
$886$	5.53987	0.186115
$887$	21.9710	0.737713	0.368857	−	0.929486i	$-0.379749\pi$
$887$	21.9710	0.737713	0.368857	+	0.929486i	$0.379749\pi$
$888$	0	0
$889$	24.1279	0.809224
$890$	12.0885	0.405206
$891$	0	0
$892$	−2.55189	−0.0854435
$893$	−63.5266	−2.12583
$894$	0	0
$895$	−20.0092	−0.668835
$896$	4.42518	0.147835
$897$	0	0
$898$	−13.5471	−0.452073
$899$	−19.2938	−0.643485
$900$	0	0
$901$	−8.75682	−0.291732
$902$	−6.74996	−0.224749
$903$	0	0
$904$	3.01462	0.100265
$905$	−7.75472	−0.257776
$906$	0	0
$907$	−47.0082	−1.56088	−0.780441	−	0.625229i	$-0.785005\pi$
$907$	−47.0082	−1.56088	−0.780441	+	0.625229i	$0.785005\pi$
$908$	−19.7228	−0.654524
$909$	0	0
$910$	−25.8403	−0.856597
$911$	−11.5694	−0.383310	−0.191655	−	0.981462i	$-0.561385\pi$
$911$	−11.5694	−0.383310	−0.191655	+	0.981462i	$0.561385\pi$
$912$	0	0
$913$	−9.13643	−0.302372
$914$	−11.1137	−0.367609
$915$	0	0
$916$	11.2054	0.370237
$917$	−38.3994	−1.26806
$918$	0	0
$919$	−35.1441	−1.15930	−0.579648	−	0.814867i	$-0.696810\pi$
$919$	−35.1441	−1.15930	−0.579648	+	0.814867i	$0.696810\pi$
$920$	0	0
$921$	0	0
$922$	−33.3324	−1.09774
$923$	28.8008	0.947991
$924$	0	0
$925$	−8.53541	−0.280642
$926$	19.4351	0.638679
$927$	0	0
$928$	−6.39946	−0.210073
$929$	35.7727	1.17366	0.586832	−	0.809709i	$-0.300375\pi$
$929$	35.7727	1.17366	0.586832	+	0.809709i	$0.300375\pi$
$930$	0	0
$931$	−83.0615	−2.72223
$932$	−18.0191	−0.590236
$933$	0	0
$934$	33.5250	1.09697
$935$	6.91265	0.226068
$936$	0	0
$937$	−17.6705	−0.577270	−0.288635	−	0.957439i	$-0.593201\pi$
$937$	−17.6705	−0.577270	−0.288635	+	0.957439i	$0.593201\pi$
$938$	−53.3027	−1.74039
$939$	0	0
$940$	−10.2708	−0.334996
$941$	−15.9220	−0.519041	−0.259521	−	0.965738i	$-0.583565\pi$
$941$	−15.9220	−0.519041	−0.259521	+	0.965738i	$0.583565\pi$
$942$	0	0
$943$	0	0
$944$	−9.75208	−0.317403
$945$	0	0
$946$	4.80582	0.156251
$947$	19.3751	0.629606	0.314803	−	0.949157i	$-0.398062\pi$
$947$	19.3751	0.629606	0.314803	+	0.949157i	$0.398062\pi$
$948$	0	0
$949$	39.3131	1.27616
$950$	−25.4874	−0.826920
$951$	0	0
$952$	17.3738	0.563089
$953$	−29.5147	−0.956073	−0.478037	−	0.878340i	$-0.658651\pi$
$953$	−29.5147	−0.956073	−0.478037	+	0.878340i	$0.658651\pi$
$954$	0	0
$955$	−26.8766	−0.869706
$956$	−27.2479	−0.881261
$957$	0	0
$958$	1.65573	0.0534941
$959$	−39.7459	−1.28346
$960$	0	0
$961$	−21.9103	−0.706784
$962$	−12.0953	−0.389969
$963$	0	0
$964$	22.2765	0.717478
$965$	−18.0447	−0.580879
$966$	0	0
$967$	5.25882	0.169112	0.0845562	−	0.996419i	$-0.473053\pi$
$967$	5.25882	0.169112	0.0845562	+	0.996419i	$0.473053\pi$
$968$	8.27868	0.266087
$969$	0	0
$970$	−12.7857	−0.410525
$971$	20.5102	0.658204	0.329102	−	0.944294i	$-0.393254\pi$
$971$	20.5102	0.658204	0.329102	+	0.944294i	$0.393254\pi$
$972$	0	0
$973$	−13.5538	−0.434513
$974$	20.9954	0.672737
$975$	0	0
$976$	1.55946	0.0499169
$977$	1.74232	0.0557418	0.0278709	−	0.999612i	$-0.491127\pi$
$977$	1.74232	0.0557418	0.0278709	+	0.999612i	$0.491127\pi$
$978$	0	0
$979$	−18.6840	−0.597145
$980$	−13.4291	−0.428978
$981$	0	0
$982$	25.8128	0.823720
$983$	0.0235072	0.000749763 0	0.000374881	−	1.00000i	$-0.499881\pi$
$983$	0.0235072	0.000749763 0	0.000374881		1.00000i	$0.499881\pi$
$984$	0	0
$985$	−13.7063	−0.436720
$986$	−25.1251	−0.800146
$987$	0	0
$988$	−36.1176	−1.14905
$989$	0	0
$990$	0	0
$991$	−15.8169	−0.502441	−0.251220	−	0.967930i	$-0.580832\pi$
$991$	−15.8169	−0.502441	−0.251220	+	0.967930i	$0.580832\pi$
$992$	3.01491	0.0957236
$993$	0	0
$994$	23.2948	0.738868
$995$	14.5203	0.460324
$996$	0	0
$997$	−9.79242	−0.310129	−0.155065	−	0.987904i	$-0.549559\pi$
$997$	−9.79242	−0.310129	−0.155065	+	0.987904i	$0.549559\pi$
$998$	6.43766	0.203780
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.br.1.2		5
3.2	odd	2		3174.2.a.bb.1.4		5
23.7	odd	22		414.2.i.e.325.1		10
23.10	odd	22		414.2.i.e.307.1		10
23.22	odd	2		9522.2.a.bs.1.4		5
69.53	even	22		138.2.e.b.49.1	yes	10
69.56	even	22		138.2.e.b.31.1	✓	10
69.68	even	2		3174.2.a.ba.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.b.31.1	✓	10	69.56	even	22
138.2.e.b.49.1	yes	10	69.53	even	22
414.2.i.e.307.1		10	23.10	odd	22
414.2.i.e.325.1		10	23.7	odd	22
3174.2.a.ba.1.2		5	69.68	even	2
3174.2.a.bb.1.4		5	3.2	odd	2
9522.2.a.br.1.2		5	1.1	even	1	trivial
9522.2.a.bs.1.4		5	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.06731 q^{5} -4.42518 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.06731 q^{5} -4.42518 q^{7} -1.00000 q^{8} +1.06731 q^{10} -1.64964 q^{11} +5.47112 q^{13} +4.42518 q^{14} +1.00000 q^{16} +3.92613 q^{17} -6.60149 q^{19} -1.06731 q^{20} +1.64964 q^{22} -3.86085 q^{25} -5.47112 q^{26} -4.42518 q^{28} +6.39946 q^{29} -3.01491 q^{31} -1.00000 q^{32} -3.92613 q^{34} +4.72304 q^{35} +2.21076 q^{37} +6.60149 q^{38} +1.06731 q^{40} -4.09177 q^{41} +2.91325 q^{43} -1.64964 q^{44} +9.62306 q^{47} +12.5822 q^{49} +3.86085 q^{50} +5.47112 q^{52} -2.23040 q^{53} +1.76068 q^{55} +4.42518 q^{56} -6.39946 q^{58} -9.75208 q^{59} +1.55946 q^{61} +3.01491 q^{62} +1.00000 q^{64} -5.83938 q^{65} -12.0453 q^{67} +3.92613 q^{68} -4.72304 q^{70} +5.26416 q^{71} +7.18556 q^{73} -2.21076 q^{74} -6.60149 q^{76} +7.29997 q^{77} -2.69381 q^{79} -1.06731 q^{80} +4.09177 q^{82} +5.53843 q^{83} -4.19039 q^{85} -2.91325 q^{86} +1.64964 q^{88} +11.3261 q^{89} -24.2107 q^{91} -9.62306 q^{94} +7.04583 q^{95} -11.9794 q^{97} -12.5822 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - q^{5} - 11 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 - q^5 - 11 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} - q^{5} - 11 q^{7} - 5 q^{8} + q^{10} + 11 q^{11} + 12 q^{13} + 11 q^{14} + 5 q^{16} - q^{17} - 15 q^{19} - q^{20} - 11 q^{22} + 6 q^{25} - 12 q^{26} - 11 q^{28} - q^{29} - 18 q^{31} - 5 q^{32} + q^{34} + 11 q^{35} - 10 q^{37} + 15 q^{38} + q^{40} + 16 q^{41} - 18 q^{43} + 11 q^{44} + 4 q^{47} + 20 q^{49} - 6 q^{50} + 12 q^{52} - q^{53} - 22 q^{55} + 11 q^{56} + q^{58} - 2 q^{59} + q^{61} + 18 q^{62} + 5 q^{64} + 24 q^{65} - 29 q^{67} - q^{68} - 11 q^{70} + 11 q^{71} - 8 q^{73} + 10 q^{74} - 15 q^{76} - 11 q^{77} - 40 q^{79} - q^{80} - 16 q^{82} + 8 q^{83} - 13 q^{85} + 18 q^{86} - 11 q^{88} + 2 q^{89} - 33 q^{91} - 4 q^{94} + 3 q^{95} - 17 q^{97} - 20 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - q^5 - 11 * q^7 - 5 * q^8 + q^10 + 11 * q^11 + 12 * q^13 + 11 * q^14 + 5 * q^16 - q^17 - 15 * q^19 - q^20 - 11 * q^22 + 6 * q^25 - 12 * q^26 - 11 * q^28 - q^29 - 18 * q^31 - 5 * q^32 + q^34 + 11 * q^35 - 10 * q^37 + 15 * q^38 + q^40 + 16 * q^41 - 18 * q^43 + 11 * q^44 + 4 * q^47 + 20 * q^49 - 6 * q^50 + 12 * q^52 - q^53 - 22 * q^55 + 11 * q^56 + q^58 - 2 * q^59 + q^61 + 18 * q^62 + 5 * q^64 + 24 * q^65 - 29 * q^67 - q^68 - 11 * q^70 + 11 * q^71 - 8 * q^73 + 10 * q^74 - 15 * q^76 - 11 * q^77 - 40 * q^79 - q^80 - 16 * q^82 + 8 * q^83 - 13 * q^85 + 18 * q^86 - 11 * q^88 + 2 * q^89 - 33 * q^91 - 4 * q^94 + 3 * q^95 - 17 * q^97 - 20 * q^98

Introduction

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